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Aug
21
comment Additivity of upper densities with respect to arithmetic progressions of integers
I've just realized that the answer to my previous question is negative: Looking at the case of the upper asymptotic density, ${\sf d}^\ast$, on $\mathbf N^+$, fix $\alpha \in {]0,1[}$ and consider as $X$ the set $\bigcup_{n \ge 1} [\![\alpha(2n-1)! + (1-\alpha)(2n)!+1, (2n)!]\!]$; then recall from mathoverflow.net/questions/207522 that ${\sf d}^\ast(X) = \alpha$. Edit: I see we added a comment almost at the same time.
Aug
21
comment Additivity of upper densities with respect to arithmetic progressions of integers
I didn't ask you to agree with me (on something); I asked if you have a counterexample (to something). Anyway, here is an idea: Let $\mathcal A^\sharp$ be the collection of all subsets of $\mathbf N^+$ that can be expressed as a finite union of sets of arithmetic progressions of ${\bf N}^+$, or which differ from these by finitely many integers. Given $X\subseteq{\bf N}^+$, can you show that there exists a nonincreasing (resp., nondecreasing) sequence $(U_n)_{n \ge 1}$ of $\mathcal A^\sharp$ such that $X\subseteq U_n$ (resp., $U_n\subseteq X$) for each $n$ and $\lim_n\mu^\ast(U_n)=\mu^\ast(X)$?
Aug
18
comment Additivity of upper densities with respect to arithmetic progressions of integers
Do you have any counterexample to the stronger (and more natural?) statement that $\mu^\ast(X \cup Y) = \mu^\ast(X) + \mu^\ast(Y)$ whenever $X,Y \subseteq \mathbf N^+$, $X \cap Y = \emptyset$, and $\mu^\ast(Y) + \mu^\ast(Y^c) = 1$, i.e. $Y$ belongs to the domain of the density induced by $\mu^\ast$?
Aug
15
revised On the independence of lower and upper asymptotic and Banach densities
Fixed a typo in the definitions of the upper and lower Banach densities
Aug
15
comment On the independence of lower and upper asymptotic and Banach densities
Ops, that's a typo! Thanks for pointing it out, I'm going to edit.
Jul
21
comment Terminology for torsion semigroups where the order of elements is uniformly finite
I was looking forward to your comment, @BenjaminSteinberg, and will probably follow your suggestion, although I'm slightly puzzled by the fact that neither MathSciNet nor Google seem to credit it much, up to few remarkable positive feedbacks (e.g., J. Rhodes has used "Bounded torsion semigroups" in Infinite Iteration of Matrix Semigroups I. Structure Theorem for Torsion Semigroups, J. Algebra 98 (Feb 1986), No. 2, 422–451).
Jul
20
comment Terminology for torsion semigroups where the order of elements is uniformly finite
Any (sufficiently strong) evidence in support of your believes?
Jul
20
revised Terminology for torsion semigroups where the order of elements is uniformly finite
Fixed a typo in the title
Jul
20
asked Terminology for torsion semigroups where the order of elements is uniformly finite
Jul
19
revised Distribution of residue classes of totients of (univariate) polynomials
Added a more specific question and some explanations
Jul
19
comment Distribution of residue classes of totients of (univariate) polynomials
I doubt it too, but special cases (I mean, cases corresponding to some specific choice of $f$) would already be interesting to me. E.g., what about the linear/quadratic case? (Btw, thanks for the link.)
Jul
19
revised Distribution of residue classes of totients of (univariate) polynomials
Removed a pointless assumption
Jul
19
asked Distribution of residue classes of totients of (univariate) polynomials
Jul
18
awarded  Yearling
Jul
11
comment On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
@NikitaSidorov: Do you still think the same, after the edit? If so, I'm interested. Btw, the scenario considered in the OP is not really the most general one that I've in mind, but should be already large enough to rule out any approach purely based on the analysis of the asymptotic behavior of the counting fnc of $X$ (as per Anthony Quas' answer below).
Jul
11
comment On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
I've just given it a try.
Jul
11
revised On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
Fixed a typo in the title and hopefully clarified the question
Jul
11
revised On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
Fixed a typo in the title and hopefully clarified the question
Jul
10
comment Suprema of lower density of sums and products of sets with lower density 0
I don't get the above comments. (i) What does $A \cup B$ have to do with $A+B$? Maybe $A = \{0\} \cup \bigcup_{i \ge 1} [\![n_{2i-1},n_{2i}]\!]$? To avoid misunderstandings: I'm asking about the presence of zero in $A$, not about the rest. (ii) It is unclear to me whether the OP includes $0$ in $\mathbf N$, which makes a difference with Anthony Quas' construction. In any case, the first question is essentially a duplicate of mathoverflow.net/questions/205843, where it is shown that the answer is 1 (no matter whether or not $0 \in \mathbf N$).
Jul
10
comment Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?
In addition, the full reference to the paper of Kronecker mentioned in the above comments is: L. Kronecker, Quelques remarques sur la détermination des valeurs moyennes, C. R. Math. Acad. Sci. Paris 103 (Nov., 1886), 980-987. The paper is available to download from the website of the Bibliotèque nationale de France (gallica.bnf.fr/ark:/12148/cb343481087/date.r=.langEN).